The theory of determinants in the historical order of development, by Sir Thomas Muir.

46 HISTORY OF THE THEORY OF DETERMINANTS

in which any member of any set is a linear homogeneous function
of all the members of any other set. Of course, it is impossible
that more than three connecting sets of equations can be independent. For example, if it be given that
(Y, Y2, Y 3)= ( 11 12 13 6 x1, x2, X3), (1, 172, 713) ( a1 a2 a3 iYi,, Y2 y3)
ml % ms m^ \ 66
MI m2 m3 b b2 b3
n ln2 n3 I 1C C2 C3
(~, ~', 713) = ( k k2 X3 ', 2, &3), (1, 2, 3)=(f f2f3 X1, x2, X3)
rx ~J2 r8T glg2g3
V1 1'2 V3 hI h h3
it is evident that there must exist between the coefficients the
matrical relation

( a,
b,
C,

a2 a3 1
b2 b3 ml
c2 C3 ~ i1

2 13 )
M?2 in3
n2 n3
m^^ ~3

= ( X1 X 2 3 fAA f 3 )
Ml 112 113 1 gh2 g3
VI V2 V3 |I h 2 h2 3.

Now, what Hamburger in effect says-he does not use Cayley's
notation *-is that, if the third matrix in this last identity be the
same as the second, then

a1 -b,,

a2 a3
b2 — b3
C2 c3-0

fi-e
91
h,

f2-0
h,

f3
g9
h3-0

for all values of 0. By way of proof it is pointed out that in this
special case the matrical relation becomes

( a a2 a3 al a2 a3 a a a3 )
11 m1 1 12 m2 n2 13 m3n3
b, b b3 b1 b b2 bb2 b b3
1x m nl 12 m2 n2 13 m 3n
c1 c2 c3 cl c2 C3 c1 c2 03
li m1 n1 12 m2 n2 13 m3 n
* See Hist., i. pp. 85-87.

_ ( 12 1, 3
fi g1 h,
mI m2 m3
fi gI h1
n1 n2 n3
fi g1 h1

'l 12 13 1 1 2 l. )
f2 g2 h2 f3 g3 h3
f2 g2 i2 f3 g3 I&.
nl n2 n3 n, t2 n3
f g2 h2 f3 g3 h.3,